A Simulation-Based Scaled Test Statistic for Assessing Model-Data Fit in Least-Squares Unrestricted Factor-Analysis Solutions

A shortcoming of least-squares unrestricted factor analysis (UFA) procedures, which are widely used in psychometric applications is that a test statistic for assessing model-data fit cannot be easily derived from the minimum fit function value. This paper proposes a chi-square type goodness-of-fit test statistic intended for the principal-axis, MINRES, and minimum-rank UFA procedures. The statistic is empirically obtained via intensive simulation based on a two-stage approach. First, a distribution of minimum fit function values is obtained from a scenario in which the null hypothesis of perfect model-data fit holds. Second, the obtained statistic is non-linearly transformed so that it has its first four moments equal to those of the theoretical reference chisquare distribution with the appropriate degrees of freedom. Extensions of the basic statistic are next proposed that include comparative and relative indexes based on it. Tests of close-fit and power assessment derived from the basic statistic are also proposed.